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ENERGY
195
H
where k is a constant and r is the distance of the body from the center o the earth. But at the surface of the earth the weight of the body is — mg, therefore F = —mg when r = a, where a is the radius of the earth. Therefore making these substitutions in the last equation we obtain
k
— mg = —, or k = — mga2. a"
Therefore and
o Crdr
= - mga~ I   -Ja r2
mga       mga
9
FIG. 103.
DISCUSSION. — Plotting the potential energy as abscissa and th height above the surface of the earth as ordinate we obtain the curve c Fig. 103, where the circle represents the earth.
When r = a, U ~ 0; as it should. When r = GO , U = mga. Therefor mga is the maximum value of the potential energy. In the figure this i evident from the fact that the curve approaches asymptotically to th
line U = mga.
When r= 2 a, U = ~jp. 2
Therefore at a height of abou
4000 miles the potential energy equals half its maximum value.
It will be seen from the following analysis that for small heights th potential energy may be considered to increase linearly with h, where h i the height above the surface of the earth:
U = mga*(- - -} \a     rJ
Jl____1_\
\a     a+h}
• mga*
* The symbols "<C" and "] Thus "h^Z. a" should be read
= mgh,   when   h <C a.*
]^>" will be used to denote great inequalities h is negligible compared with a" or "h i
very small compared with a."    On the other hand "a "a is very large compared with A."
should be rea